F equals ma and PDE. So this is the Lagrangian acceleration of a particle equals the force,

which is the pressure that maintains incompressibility. Incompressibility is the constraint that the

divergence of this vector field is zero for all time. That completely determines the pressure

of that constant. And finally, the fluids we'll consider will be non-penetrating, which

means that the velocity is just tangent to the boundaries of the domain. Now, we'll use

some simplifications that happen in 2D throughout the talk. So one of them is that since you

have incompressibility, you can introduce a stream function, which is the skew gradient

of some scalar function, c, such that the velocity is a skew gradient. So, namely, what

the stream function tells you is that level sets of this function, c, are streamlines

in that the velocity is tangent to them. The velocity is just circulating around level

sets of the stream function. And the boundary condition that the velocity is parallel to

the boundary just becomes that the stream function is constant on the connected components

of the boundary. Another simplifying structure in two dimensions is that the vorticity or

the curl of the velocity only has one component if you regard this 2D field as a 3D one with

symmetry. So it's just pointing up out of the plane that the velocity lives in. And

this pre-factor, this scalar function, could be called the 2D vorticity, which is just

the skew divergence of the velocity. And the amazing property of this vorticity is that

for 2D Euler, the structure is such that this scalar function is just transported as a scalar.

So it's just being pushed around by the integral curves of u, pushed on those curves, whereas

u is recovered from the vorticity by some non-local operation called the Biot-Savart

law. So you just have to invert the curl, which amounts to this operator. And so this

PDE formulation is equivalent to saying that the vorticity is constant on characteristics

xt of the velocity field.

So we'll be interested in the long-term behavior of the system. It's formally an infinite-dimensional

Li-Poisson or Hamiltonian system, so it has various conserved quantities, and its long-term

behavior is correspondingly complicated. So the first most well-known conserved quantity,

which holds independent of dimension, is the kinetic energy, which is just the L2 norm

of the velocity. And then in two dimensions, particularly because of this special transport

structure of the vorticity, you have conservation of the so-called chasmiers. These are just

any continuous function of the vorticity integrated over the domain. And both of these points

just use the fact that u is tangent to the boundary and incompressible to derive them.

So a special case of this latter one is that all Lp norms of the vorticity are preserved

for all time, including an infinity norm, since it's just transported as a scalar.

Now, there can be additional invariants besides energy and chasmiers, so there could be, for

example, linear momentum if your domain has some special symmetries. So this is preserved

if the domain is a periodic box, a tutors, or if it's a periodic channel, or there's

an analog on the disk. But in general, having this type of invariant requires more structure,

and it doesn't hold on a general domain. So this will play less of a role in our analysis.

So the first point is that preservation of the Lp norms of vorticity allow you to prove

this classical theorem, which says that the 2D Euler equations are globally well posed

in holder spaces for the vorticity. So namely, given any C alpha initial vorticity field,

there's a unique solution that's C alpha spacetime for all time. And moreover, you can prove

this bound on the solution. It says that the C alpha norm grows at most double exponentially.

So the dependence on time is here. And what this is saying in a way is that C alpha is

a good phase space for the Euler equation for all time, for all finite times, but at

infinite time, you can leave the space. You can, in principle, you could grow at worse

like this bound, and there are examples that actually show vorticity growth with this rate

sharply. So at infinite time, you leave the C alpha space, but at all finite times, it's

a good phase space. However, because you leave at infinite time, this is often, I mean, this

is not a very convenient way to describe what you see in the fluid at the late stage. And